Formulas for non-holomorphic Eisenstein series and for the Riemann zeta function at odd integers

TL;DR

This paper derives new Fourier expansion formulas for non-holomorphic Eisenstein series, enabling non-holomorphic analogs of classical formulas involving Eichler integrals, with implications for understanding the Riemann zeta function at odd integers.

Contribution

It introduces new explicit formulas for non-holomorphic Eisenstein series and their Fourier expansions, extending classical results to a non-holomorphic setting.

Findings

New Fourier expansion formulas for non-holomorphic Eisenstein series.

Non-holomorphic analogs of Ramanujan, Grosswald, and Berndt formulas.

Applications to the Riemann zeta function at odd integers.

Abstract

New expressions are given for the Fourier expansions of non-holomorphic Eisenstein series with weight $k$. Among other applications, this leads to non-holomorphic analogs of formulas of Ramanujan, Grosswald and Berndt containing Eichler integrals of holomorphic Eisenstein series.